Multivariate analysis of PET pharmacokinetic parameters improves inferential efficiency

Purpose In positron emission tomography quantification, multiple pharmacokinetic parameters are typically estimated from each time activity curve. Conventionally all but the parameter of interest are discarded before performing subsequent statistical analysis. However, we assert that these discarded parameters also contain relevant information which can be exploited to improve the precision and power of statistical analyses on the parameter of interest. Properly taking this into account can thereby draw more informative conclusions without collecting more data. Methods By applying a hierarchical multifactor multivariate Bayesian approach, all estimated parameters from all regions can be analysed at once. We refer to this method as Parameters undergoing Multivariate Bayesian Analysis (PuMBA). We simulated patient–control studies with different radioligands, varying sample sizes and measurement error to explore its performance, comparing the precision, statistical power, false positive rate and bias of estimated group differences relative to univariate analysis methods. Results We show that PuMBA improves the statistical power for all examined applications relative to univariate methods without increasing the false positive rate. PuMBA improves the precision of effect size estimation, and reduces the variation of these estimates between simulated samples. Furthermore, we show that PuMBA yields performance improvements even in the presence of substantial measurement error. Remarkably, owing to its ability to leverage information shared between pharmacokinetic parameters, PuMBA even shows greater power than conventional univariate analysis of the true binding values from which the parameters were simulated. Across all applications, PuMBA exhibited a small degree of bias in the estimated outcomes; however, this was small relative to the variation in estimated outcomes between simulated datasets. Conclusion PuMBA improves the precision and power of statistical analysis of PET data without requiring the collection of additional measurements. This makes it possible to study new research questions in both new and previously collected data. PuMBA therefore holds great promise for the field of PET imaging.


Supplementary Materials S1 : Prior Specification Global Intercepts
Below are the priors defined for the global intercepts. Note that all priors are defined over the natural logarithms of the parameters. The priors for α K1 and α BP P are defined for the dorsolateral prefrontal cortex as the reference level of the dummy variable. The priors for α K1 and α BP P are defined for the dorsolateral prefrontal cortex as the reference level of the dummy variable.

One-Tissue Compartment Model
The priors for α K1 and α V T are defined for the anterior cingulate cortex as the reference level of the dummy variable. The priors for α K1 and α V T are defined for the dorsolateral prefrontal cortex as the reference level of the dummy variable.

Simplified Reference Tissue Model
The priors for α R1 and α BP ND are defined for the anterior cingulate cortex as the reference level of the dummy variable.
The priors for α R1 and α BP ND are defined for the dorsolateral prefrontal cortex as the reference level of the dummy variable.

Individual deviations
Differences between individuals were defined by specifying the primary pharmacokinetic parameters in one variance-covariance matrix.

Two-Tissue Compartment Model
For both [ 11 C]WAY100635 and [ 11 C]ABP688, we used the same priors.

One-Tissue Compartment Model
For both [ 11 C]DASB and [ 11 C]GR103545, we used the same priors.

Simplified Reference Tissue Model
For both [ 11 C]DASB and [ 11 C]WAY100635, we used the same priors.

Regional deviations
For logBP ND and logK 1 , regional differences were defined as unpooled effects using a dummy (indicator) variable defined with reference to the dorsolateral prefrontal cortex. For simplicity, all regional differences (with the exception of [ 11 C]DASB) were defined as zero-centred regularising priors with the same SD.
For [ 11 C]DASB, the same standard deviations were used, but with altered means for regions whose mean binding values were very different from the other regions. For SRTM, the mean values for $$ were defined as follows: • Insula: 0.5 • Amygdala, dorsal putamen, thalamus, ventral striatum: 1 • Midbrain: 1.5 For the 1TC, the means were defined as follows: • Dorsal putamen, ventral striatum: 0.5 • Midbrain: 1 For the remaining parameters, regional differences were defined as pooled variables, arising from a common distribution

Two-Tissue Compartment Model
For both [ 11 C]WAY100635 and [ 11 C]ABP688, we used the same priors.

One-Tissue Compartment Model
For both [ 11 C]DASB and [ 11 C]GR103545, we used the same priors.

Simplified Reference Tissue Model
For both [ 11 C]DASB and [ 11 C]WAY100635, we used the same priors.

Covariates
For the simulations, there was only one additional covariate for group effects on the binding parameter. For this, we used a zero-centred regularising prior. This prior conservatively assumes that no difference is most likely, and assigns 95% of its probability between differences of -48% and 48% differences between groups.

Supplementary Materials S2 : Simulation Parameters
Below are presented the values of the parameters described in the section above in order to simulate new data. We used the posterior mean values estimated using empirical data for each application.

Regional Deviations
Regional deviations are the deviations from the global intercept for each parameter in each region. The values will be zero for the dummy variable when fixed effects were used by definition.

Covariance Matrices
For the covariance matrices, the correlations and standard deviations have been separated for the purposes of visualisation and interpretation. The correlation matrices are shown above, with the relevant standard deviations presented in the table below. For the 2TC, we show the correlation matrix for the individual variation and region-within-individual variation as these parameters are estimated using SiMBA-derived parameters. For the 1TC and SRTM, we show the PuMBA-derived correlation matrices representing individual (τ ) variation and residual (ϵ) variation.

Two-Tissue Compartment Model
[ 11 C]WAY100635. Note that these parameters are generated using SiMBA.

Supplementary Materials S4 : Comparison with SiMBA Additional Figures
Here we show additional outcomes from the TAC simulations in which PuMBA was compared with SiMBA. These figures show the false positive rate and standard deviation of estimates. Please refer to the manuscript for how each of these metrics are defined.

Supplementary Materials S5 : Correlated and Uncorrelated Data
To assess correlation matrix recovery, we simulated both both parameters as well as TACs with the same mean and variance for all parameters, but using univariate instead of multivariate distributions, i.e. with no correlation structure between the parameters, for the individual (τ ) and residual (ϵ) distributions in the parameter simulations, and for the individual and individual × region distributions for the TAC simulations using SiMBA. We generated 500 simulated datasets for each condition.
For the TAC simulations, residual correlation matrices for each tracer were very similar for correlated and uncorrelated data. This suggests that the correlations in the residual variation (ϵ) are primarily attributable to errors introduced during TAC fitting using NLS to generate the PK parameters owing to imperfect identifiability. Furthermore, comparing sample sizes of n=20, n=50 and n=100 per group, we observe broadly similar results, but with increased sample-to-sample variation in the smaller samples, suggesting that the poor parameter recovery is more attributable to NLS TAC fitting than to insufficient sample sizes. These residual correlation matrices differed greatly between tracers, suggesting that the nature of the imperfect identifiability is tracer-dependent. In the individual (τ ) correlation matrices, there were clear differences between the estimated correlation matrices for the correlated and uncorrelated datasets for each tracer, suggesting that the correlation structure of the data does affect the estimated multivariate correlation structure even if the recovery of the individual correlation coefficients themselves is poor. As shown below, in uncorrelated data relative to correlated data, false positive rates are unchanged, however power is reduced and standard error and standard deviation are increased.

Supplementary Materials S7 : Parameter Simulation Additional Figures
Here we show the additional figures regarding the outcomes of the parameter simulations, representing the false positive rate, the mean estimated differences and, the standard deviation across datasets. Please refer to the manuscript for how each of these metrics are defined.